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Interpolation and compactness in categories of pre-institutions. (English) Zbl 0860.03035
The relationship between compactness, Craig interpolation and Robinson joint consistency were investigated by Makowsky, Shelah, as well as by the reviewer, and others, in the late seventies and in the eighties. The book: Model-theoretic logics (1985; Zbl 0587.03001 and Zbl 0587.03002) (edited by J. Barwise and S. Feferman) gives a comprehensive overview of the subject. An even more general set-up was introduced by the reviewer in his paper “A generalization of abstract model theory” [Fundam. Math. 124, 1-25 (1984; Zbl 0595.03039)]. In the paper under review the authors further analyze compactness, interpolation and other related model-theoretic properties, working in the context of pre-institutions. The latter were introduced in an earlier paper by the same authors, generalizing the notion of institution, due to Goguen and Burstall. Categories of pre-institutions allow a model-free generalization of the above mentioned notions. For their analysis the authors introduce appropriate analogues of ultraproducts, and prove a generalization of Łos’ theorem.
Reviewer: D.Mundici (Milano)

MSC:
03C95 Abstract model theory
03G30 Categorical logic, topoi
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[1] DOI: 10.1016/S0049-237X(08)70132-0 · doi:10.1016/S0049-237X(08)70132-0
[2] Mossakowski, Algebraic Methodology and Software Technology (AMAST’93) Workshops in Computing pp 137– (1994) · doi:10.1007/978-1-4471-3227-1_12
[3] DOI: 10.1016/0304-3975(90)90118-2 · Zbl 0716.03022 · doi:10.1016/0304-3975(90)90118-2
[4] Ebbinghaus, Mathematical Logic (1984)
[5] Ebbinghaus, Model-Theoretic Logics pp 25– (1985)
[6] Diaconescu, Logical Environments pp 83– (1993)
[7] Bernot, Algebraic Methodology and Software Technology (AMAST’93) Workshops in Computing pp 216– (1992)
[8] Astesiano, Springer-Verlag Lecture Notes in Computer Science 665 pp 126– (1993) · doi:10.1007/3-540-56379-2_37
[9] Tarlecki, Springer-Verlag Lecture Notes in Computer Science 240 pp 334– (1986) · doi:10.1007/3-540-17162-2_132
[10] Andreéka, Colloquia Mathematics Societatis János Bolyai 29 pp 13– (1981)
[11] DOI: 10.1016/0304-3975(85)90094-5 · Zbl 0608.68014 · doi:10.1016/0304-3975(85)90094-5
[12] DOI: 10.1016/0890-5401(88)90008-9 · Zbl 0654.68017 · doi:10.1016/0890-5401(88)90008-9
[13] Makowski, Model-Theoretic Logics pp 717– (1985)
[14] Makowski, Model-Theoretic Logics pp 645– (1985)
[15] Mac Lane, Categories for the Working Mathematician pp 5– (1971) · doi:10.1007/978-1-4612-9839-7
[16] Los, Mathematical Interpretation of Formal Systems, Studies in Logic and the Foundations of Mathematics pp 98– (1954)
[17] Goguen, J. Assoc. Comput. Mack 39 pp 95– (1992) · Zbl 0799.68134 · doi:10.1145/147508.147524
[18] Goguen, Springer-Verlag Lecture Notes in Computer Science 240 pp 313– (1986) · doi:10.1007/3-540-17162-2_131
[19] Goguen, Springer-Verlag Lecture Notes in Computer Science 164 pp 221– (1984) · doi:10.1007/3-540-12896-4_366
[20] Ehrig, Algebraic Methodology and Software Technology (AMAST’93) Workshops in Computing pp 145– (1992)
[21] Salibra, Algebraic Methods in Logic and in Computer Science pp 67– (1993)
[22] Salibra, Springer-Verlag Lecture Notes in Computer Science 665 pp 310– (1993) · doi:10.1007/3-540-56379-2_47
[23] DOI: 10.1007/BF01190411 · Zbl 0756.08005 · doi:10.1007/BF01190411
[24] Nivela, Springer-Verlag Lecture Notes in Computer Science 393 pp 220– (1989) · doi:10.1007/3-540-51722-7_13
[25] Nivela, Springer-Verlag Lecture Notes in Computer Science 332 pp 184– (1988) · doi:10.1007/3-540-50325-0_10
[26] Manca, The Unified Computation Laboratory: Modelling, Specifications and Tools pp 85– (1992)
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